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| Description: The composition of two functions is a function. Exercise 29 of [TakeutiZaring] p. 25. (The proof was shortened by Andrew Salmon, 17-Sep-2011.) |
| Ref | Expression |
|---|---|
| funco |
|
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | moexexv 1842 |
. . . . . 6
| |
| 2 | 1 | ancoms 484 |
. . . . 5
|
| 3 | funmo 4437 |
. . . . . 6
| |
| 4 | 3 | 19.21aiv 1664 |
. . . . 5
|
| 5 | funmo 4437 |
. . . . 5
| |
| 6 | 2, 4, 5 | syl2an 503 |
. . . 4
|
| 7 | 6 | 19.21aiv 1664 |
. . 3
|
| 8 | funopab 4455 |
. . 3
| |
| 9 | 7, 8 | sylibr 217 |
. 2
|
| 10 | df-co 4003 |
. . 3
| |
| 11 | 10 | funeqi 4442 |
. 2
|
| 12 | 9, 11 | sylibr 217 |
1
|
| Colors of variables: wff set class |
| Syntax hints: |
| This theorem is referenced by: fnco 4521 fcoOLD 4574 f1co 4612 fvco 4736 curry1 5075 curry2 5078 mapenlem1 5583 vsfval 9586 domrancur1b 14548 domrancur1c 14550 f1ocan1fv 15717 |
| This theorem was proved from axioms: ax-1 4 ax-2 5 ax-3 6 ax-mp 7 ax-7 1304 ax-gen 1305 ax-8 1306 ax-9 1307 ax-10 1308 ax-11 1309 ax-12 1310 ax-14 1312 ax-17 1317 ax-4 1319 ax-5o 1321 ax-6o 1324 ax-9o 1481 ax-10o 1500 ax-16 1580 ax-11o 1588 ax-ext 1865 ax-sep 3438 ax-nul 3445 ax-pow 3481 ax-pr 3524 |
| This theorem depends on definitions: df-bi 164 df-or 241 df-an 242 df-ex 1327 df-sb 1536 df-eu 1775 df-mo 1776 df-clab 1872 df-cleq 1877 df-clel 1880 df-ne 2019 df-v 2294 df-dif 2597 df-un 2600 df-in 2603 df-ss 2605 df-nul 2876 df-pw 3035 df-sn 3049 df-pr 3050 df-op 3053 df-br 3339 df-opab 3396 df-id 3586 df-xp 4000 df-rel 4001 df-cnv 4002 df-co 4003 df-fun 4008 |